Nicolai Reshetikhin

The Knizhnik-Zamolodchikov system as a deformation of the Isomonodromy problem

It is shown that the Knizhnik-Zamolodchikov (KZ) equation (and corresponding vector bundle) can be viewed as a quantization of the isomonodromy problem for differential equations with several singular points.

Dimers on Surface Graphs and Spin Structures. II

In a previous paper [3], we showed how certain orientations of the edges of a graph Γ embedded in a closed oriented surface Σ can be understood as discrete spin structures on Σ. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on Γ. In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function. These operations allow to reformulate the dimer model as a quantum field theory on surface graphs.

Jackson-type integrals, bethe vectors, and solutions to a difference analog of the Knizhnik-Zamolodchikov system

It is shown how to find solutions to the quantum Knizhnik-Zamolodchikov system using the Bethe ansatz technique.

Classical BV Theories on Manifolds with Boundary

In this paper we extend the classical BV framework to gauge theories on spacetime manifolds with boundary. In particular, we connect the BV construction in the bulk with the BFV construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners. We present several examples including electrodynamics, Yang-Mills theory and topological field theories coming from the AKSZ construction, in particular, the Chern-Simons theory, the BF theory, and the Poisson sigma model. This paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing.

The poisson structure on the moduli space of flat connections and chord diagrams

Abstract We Introduce the notion of chord diagrams on arbitrary compact (possibly punctured) oriented surfaces. In the case of the 2-spheres these are just the usual chord diagrams used in test study of Vassiliev invariants of links. We consider the algebra of chord diagrams on a surface and prove that this algebra has a natural Poisson structure. Suppose now that G is a Lie group with an invariant bilinear form on g = Lie(G). We can associate to each chord diagram (coloured by representations of G) a function on the moduli space of flat G-connections on the surface. Our main result states that this map is a Poisson algebra homomorphism. Moreover, for most classical groups we prove that any algebraic function on moduli space can be obtained this way and we conjecture that this holds for all simple groups. In this way we obtain a universal description of the Poisson algebra of the moduli space, decoupling the Lie group in question.

Quasitriangularity of quantum groups at roots of 1

An important property of a Hopf algebra is its quasitriangularity and it is useful for various applications. This property is investigated for quantum groupssl2 at roots of 1. It is shown that different forms of the quantum groupsl2 at roots of 1 are either quasitriangular or have similar structure which will be called braiding. In the most interesting cases this property means that “braiding automorphism” is a combination of some Poisson transformation and an adjoint transformation with a certain element of the tensor square of the algebra.

Boundary Quantum Knizhnik–Zamolodchikov Equations and Bethe Vectors

Solutions to boundary quantum Knizhnik–Zamolodchikov equations are constructed as bilateral sums involving “off-shell” Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of

Boundary Quantum Knizhnik–Zamolodchikov Equations and Fusion

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